Podcast
Questions and Answers
When is the expression $\frac{51x^2 + 24x}{17x}$ undefined?
When is the expression $\frac{51x^2 + 24x}{17x}$ undefined?
- x = 0 (correct)
- x = -2
- The expression is defined for all real numbers
- x = -3/8
What values of x make the expression $\frac{24x^2 + 9x}{8x + 3}$ undefined?
What values of x make the expression $\frac{24x^2 + 9x}{8x + 3}$ undefined?
- x = -3/8 (correct)
- x = 4
- x = -4
- x = 0
What is the simplified form of $\frac{105}{42}$?
What is the simplified form of $\frac{105}{42}$?
- 1/2
- 2/3
- 7/13
- 5/2 (correct)
List the two values of x that make the expression $\frac{x^3 - 9x}{x^2 + 5x + 6}$ undefined.
List the two values of x that make the expression $\frac{x^3 - 9x}{x^2 + 5x + 6}$ undefined.
What is the result of multiplying $\frac{12}{7}$ and $\frac{49}{36}$?
What is the result of multiplying $\frac{12}{7}$ and $\frac{49}{36}$?
A rational expression is defined when its denominator is zero.
A rational expression is defined when its denominator is zero.
Match the following rational expressions with when they are undefined:
Match the following rational expressions with when they are undefined:
A basic rule of fractions states that for any rational expression $\frac{ac}{bc}$, if _____ and _____, then $\frac{ac}{bc} = \frac{a}{b}$.
A basic rule of fractions states that for any rational expression $\frac{ac}{bc}$, if _____ and _____, then $\frac{ac}{bc} = \frac{a}{b}$.
Flashcards are hidden until you start studying
Study Notes
Undefined Rational Expressions
- Rational numbers can be expressed as fractions of two integers; rational expressions are fractions of two polynomials.
- Denominators in rational expressions cannot equal zero to avoid undefined values.
- The domain of a rational expression includes all variable values except those that make the denominator zero.
- To find undefined values, set the denominator to zero and solve for the variable.
- Examples of expressions that are undefined:
- For ( \frac{51x^2 + 24x}{17x} ), undefined at ( x = 0 ).
- For ( \frac{24x^2 + 9x}{8x + 3} ), undefined at ( x = -\frac{3}{8} ).
- For ( \frac{x^3 - 9x}{x^2 + 5x + 6} ), undefined at ( x = -3, -2 ).
- For ( \frac{x^2 + 6x + 8}{x^2 - 16} ), undefined at ( x = -4, 4 ).
- The expression ( \frac{x^3 + 2x^2 + x + 2}{x^2 + 1} ) is defined for all real numbers as ( x^2 + 1 ) is never zero.
Simplifying Rational Expressions
- Simplification follows the rule: if ( \frac{ac}{bc} ) (where ( b ≠0 ) and ( c ≠0 )), then ( \frac{ac}{bc} = \frac{a}{b} ).
- To simplify, factor both the numerator and denominator, then divide out common factors.
- Example simplifications:
- ( \frac{105}{42} = \frac{5}{2} ) after factoring.
- ( \frac{14}{21} = \frac{2}{3} ).
- ( \frac{21}{39} = \frac{7}{13} ).
- ( \frac{15x}{13x} = \frac{15}{13} ).
- ( \frac{2x + 6}{3x + 9} = \frac{2}{3} ).
- ( \frac{x^2 + 9x + 14}{x^2 - 4} = \frac{x + 7}{x - 2} ).
- ( \frac{x^2 + 2x - 3}{x^2 - 5x + 4} = \frac{x + 3}{x - 4} ).
Multiplying Rational Expressions
- To multiply rational expressions, first factor the numerators and denominators completely.
- Identify and reduce common factors before multiplying.
- The Quotient Rule states ( \frac{x^a}{x^b} = x^{a-b} ) for any real numbers ( a ) and ( b ), under the condition ( x ≠0 ).
- Example of multiplication:
- ( \left(\frac{12}{7}\right)\left(\frac{49}{36}\right) = \frac{7}{3} ) after factoring and reducing.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
